BRMS models’ validation
This document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and total biomass as the focal ecosystem function.
Important terms:
Stage: With seed inflow, without seed inflowNinitial: Planted species richness
1 Grass1
Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
1.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 2.35 3.54 -4.37 9.40 1.00 2075 2352
## StageWithoutseedinflow -12.45 4.02 -20.14 -4.11 1.00 2060 2210
## StageWithseedinflow:Shannon 17.76 1.49 14.81 20.62 1.00 2054 2012
## StageWithoutseedinflow:Shannon 25.79 1.98 21.77 29.54 1.00 2092 2050
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 23.18 0.61 22.00 24.37 1.00 2778 2345
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2971205 0.02313489 0.2487609 0.34013891.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
1.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
1.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
1.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
1.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 82.86 26.87 30.08 135.37 1.00 5734 2756
## Ninitial4:StageWithseedinflow 151.47 24.19 103.90 198.20 1.00 5389 2863
## Ninitial8:StageWithseedinflow 167.07 20.20 127.09 205.43 1.00 6224 2593
## Ninitial16:StageWithseedinflow 250.92 17.44 217.08 285.36 1.00 5264 2884
## Ninitial32:StageWithseedinflow 278.74 19.99 238.57 317.27 1.00 5573 3139
## Ninitial2:StageWithoutseedinflow 22.84 10.73 1.93 44.10 1.00 5833 2782
## Ninitial4:StageWithoutseedinflow 64.19 14.89 34.84 92.98 1.00 5809 3160
## Ninitial8:StageWithoutseedinflow 89.41 14.54 60.77 117.58 1.00 6116 2713
## Ninitial16:StageWithoutseedinflow 184.97 17.40 151.48 218.40 1.00 5330 2982
## Ninitial32:StageWithoutseedinflow 188.96 23.27 141.86 235.28 1.00 5239 2681
## Ninitial2:StageWithseedinflow:Shannon -36.57 16.75 -69.24 -3.90 1.00 5746 2714
## Ninitial4:StageWithseedinflow:Shannon -54.24 11.07 -75.38 -32.49 1.00 5402 2823
## Ninitial8:StageWithseedinflow:Shannon -47.61 7.62 -62.29 -32.42 1.00 6253 2727
## Ninitial16:StageWithseedinflow:Shannon -65.03 5.92 -76.67 -53.52 1.00 5202 3000
## Ninitial32:StageWithseedinflow:Shannon -64.82 6.32 -76.99 -52.27 1.00 5541 3146
## Ninitial2:StageWithoutseedinflow:Shannon -3.25 7.21 -17.53 10.74 1.00 5541 2709
## Ninitial4:StageWithoutseedinflow:Shannon -19.14 7.64 -34.06 -4.10 1.00 5760 3188
## Ninitial8:StageWithoutseedinflow:Shannon -24.07 6.44 -36.96 -11.38 1.00 6132 2732
## Ninitial16:StageWithoutseedinflow:Shannon -52.87 7.18 -66.68 -38.79 1.00 5341 2858
## Ninitial32:StageWithoutseedinflow:Shannon -44.10 8.87 -61.66 -26.34 1.00 5269 2522
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.05 0.43 14.22 15.93 1.00 6762 2838
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.6997435 0.01090152 0.6772719 0.71920291.2.1 Posterior predictive checks
1.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
1.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
1.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
2 Grass2
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
2.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 44.58 1.57 41.59 47.66 1.00 2205 2333
## StageWithoutseedinflow 37.48 1.77 34.09 40.95 1.00 1971 2216
## StageWithseedinflow:Shannon 8.69 0.56 7.60 9.78 1.00 2172 2362
## StageWithoutseedinflow:Shannon 15.09 0.78 13.54 16.61 1.00 1968 2086
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 12.22 0.32 11.61 12.86 1.00 2922 1948
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4411902 0.01957698 0.4012542 0.47742412.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
2.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
2.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
2.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
2.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 84.80 33.70 15.99 150.50 1.00 4189 2834
## Ninitial4:StageWithseedinflow 136.13 36.43 63.04 209.12 1.00 4721 2397
## Ninitial8:StageWithseedinflow 133.23 31.32 72.85 195.25 1.00 4742 2594
## Ninitial16:StageWithseedinflow 91.96 50.76 -8.14 191.89 1.00 4183 2514
## Ninitial32:StageWithseedinflow 42.94 62.67 -80.61 166.86 1.00 4230 2509
## Ninitial2:StageWithoutseedinflow -16.58 22.66 -61.55 27.91 1.00 4407 2901
## Ninitial4:StageWithoutseedinflow 27.23 13.63 0.17 54.14 1.00 4674 2684
## Ninitial8:StageWithoutseedinflow 38.60 10.78 16.71 59.77 1.00 4472 2791
## Ninitial16:StageWithoutseedinflow 47.47 11.41 25.41 70.15 1.00 4346 2405
## Ninitial32:StageWithoutseedinflow 72.56 14.83 42.74 101.49 1.00 5203 2906
## Ninitial2:StageWithseedinflow:Shannon -16.03 20.42 -56.08 25.88 1.00 4196 2885
## Ninitial4:StageWithseedinflow:Shannon -29.84 15.80 -61.57 1.86 1.00 4693 2283
## Ninitial8:StageWithseedinflow:Shannon -20.14 10.81 -41.53 0.55 1.00 4729 2636
## Ninitial16:StageWithseedinflow:Shannon -4.55 14.33 -32.79 23.69 1.00 4188 2600
## Ninitial32:StageWithseedinflow:Shannon 8.12 14.93 -21.45 37.48 1.00 4227 2536
## Ninitial2:StageWithoutseedinflow:Shannon 46.13 14.00 18.44 73.91 1.00 4407 2985
## Ninitial4:StageWithoutseedinflow:Shannon 22.09 7.40 7.47 36.81 1.00 4722 2772
## Ninitial8:StageWithoutseedinflow:Shannon 17.72 4.88 8.00 27.67 1.00 4500 2836
## Ninitial16:StageWithoutseedinflow:Shannon 12.22 4.17 3.96 20.41 1.00 4316 2456
## Ninitial32:StageWithoutseedinflow:Shannon 3.26 4.42 -5.40 12.06 1.00 5192 2863
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.04 0.26 8.56 9.58 1.00 4826 2528
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4941592 0.0196986 0.4523918 0.53043862.2.1 Posterior predictive checks
2.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
2.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
2.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
3 Grass3
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
3.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 46.11 1.45 43.30 49.06 1.00 1873 1957
## StageWithoutseedinflow 46.00 1.44 43.27 48.83 1.00 1840 2019
## StageWithseedinflow:Shannon 2.68 0.54 1.62 3.72 1.00 1909 1824
## StageWithoutseedinflow:Shannon 2.88 0.57 1.75 3.99 1.00 1809 1885
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.96 0.25 9.48 10.47 1.00 2844 2314
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.06589133 0.01661286 0.03520367 0.10029293.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
3.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
3.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
3.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
3.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 18.41 36.96 -51.87 91.92 1.00 4129 3049
## Ninitial4:StageWithseedinflow -7.05 25.73 -58.38 43.21 1.00 3615 2618
## Ninitial8:StageWithseedinflow 4.23 26.82 -48.98 55.79 1.00 3929 2779
## Ninitial16:StageWithseedinflow 58.85 22.61 14.16 102.76 1.00 3958 2928
## Ninitial32:StageWithseedinflow 48.93 30.93 -9.71 110.08 1.00 3786 2671
## Ninitial2:StageWithoutseedinflow 64.95 6.46 51.66 77.13 1.00 3404 2608
## Ninitial4:StageWithoutseedinflow 25.40 11.37 3.01 47.62 1.00 3724 2853
## Ninitial8:StageWithoutseedinflow 37.27 11.59 14.76 60.20 1.00 3988 2779
## Ninitial16:StageWithoutseedinflow 65.16 21.11 24.61 105.65 1.00 3614 3027
## Ninitial32:StageWithoutseedinflow 78.89 32.49 15.88 144.36 1.00 3958 2597
## Ninitial2:StageWithseedinflow:Shannon 18.29 22.15 -26.03 60.63 1.00 4138 3035
## Ninitial4:StageWithseedinflow:Shannon 25.10 11.26 3.31 47.44 1.00 3609 2712
## Ninitial8:StageWithseedinflow:Shannon 17.30 9.48 -0.93 36.05 1.00 3925 2845
## Ninitial16:StageWithseedinflow:Shannon -1.06 6.70 -14.00 12.32 1.00 3971 2962
## Ninitial32:StageWithseedinflow:Shannon 2.32 8.12 -13.70 17.62 1.00 3793 2670
## Ninitial2:StageWithoutseedinflow:Shannon -10.02 4.01 -17.62 -1.76 1.00 3564 2543
## Ninitial4:StageWithoutseedinflow:Shannon 11.64 5.30 1.14 22.04 1.00 3733 2754
## Ninitial8:StageWithoutseedinflow:Shannon 6.02 4.42 -2.80 14.59 1.00 3986 2877
## Ninitial16:StageWithoutseedinflow:Shannon -3.50 6.64 -16.25 9.30 1.00 3586 2947
## Ninitial32:StageWithoutseedinflow:Shannon -5.77 9.23 -24.34 12.08 1.00 3962 2709
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 6.73 0.19 6.38 7.11 1.00 6008 3209
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2335496 0.02541694 0.1833591 0.28224913.2.1 Posterior predictive checks
3.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
3.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
3.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
4 Forest1
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
4.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 28.10 3.29 21.42 34.39 1.00 2109 1828
## StageWithoutseedinflow 30.66 3.78 23.41 38.21 1.00 2115 1956
## StageWithseedinflow:Shannon 15.01 1.97 11.22 19.03 1.00 2159 1990
## StageWithoutseedinflow:Shannon 13.86 2.78 8.43 19.19 1.00 2176 1945
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 21.89 0.55 20.81 23.01 1.00 2831 2463
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1043157 0.01906257 0.06958144 0.14350414.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
4.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
4.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
4.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
4.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 67.27 11.62 44.76 89.81 1.00 4244 2814
## Ninitial4:StageWithseedinflow 89.48 9.92 69.83 109.16 1.00 4379 2990
## Ninitial8:StageWithseedinflow 94.54 11.13 71.92 116.15 1.00 4154 2933
## Ninitial16:StageWithseedinflow 78.19 14.14 50.21 106.21 1.00 4285 2949
## Ninitial32:StageWithseedinflow 41.75 20.87 0.50 82.51 1.00 4315 2730
## Ninitial2:StageWithoutseedinflow 65.98 14.28 37.44 93.33 1.00 3789 2927
## Ninitial4:StageWithoutseedinflow 75.19 9.14 57.23 93.65 1.00 4670 3151
## Ninitial8:StageWithoutseedinflow 100.81 10.09 81.03 119.87 1.00 3594 2916
## Ninitial16:StageWithoutseedinflow 58.17 9.58 39.00 76.69 1.00 4101 2882
## Ninitial32:StageWithoutseedinflow 46.75 8.56 29.90 63.39 1.00 4599 3220
## Ninitial2:StageWithseedinflow:Shannon -24.74 9.70 -43.61 -5.58 1.00 4101 2877
## Ninitial4:StageWithseedinflow:Shannon -28.90 7.40 -43.83 -14.26 1.00 4337 2945
## Ninitial8:StageWithseedinflow:Shannon -23.11 7.18 -37.12 -8.54 1.00 4174 2819
## Ninitial16:StageWithseedinflow:Shannon -6.90 7.10 -20.97 7.25 1.00 4332 2922
## Ninitial32:StageWithseedinflow:Shannon 9.88 8.45 -6.82 26.82 1.00 4330 2710
## Ninitial2:StageWithoutseedinflow:Shannon -28.24 13.36 -54.82 -2.14 1.00 3749 2924
## Ninitial4:StageWithoutseedinflow:Shannon -22.83 7.44 -37.76 -8.22 1.00 4644 2824
## Ninitial8:StageWithoutseedinflow:Shannon -33.92 7.55 -48.42 -19.10 1.00 3575 3007
## Ninitial16:StageWithoutseedinflow:Shannon 1.87 6.11 -10.01 14.08 1.00 4080 3018
## Ninitial32:StageWithoutseedinflow:Shannon 8.25 4.77 -0.98 17.81 1.00 4332 2988
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 17.66 0.52 16.67 18.73 1.00 7764 2080
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.3183385 0.02441864 0.2684945 0.36377144.2.1 Posterior predictive checks
4.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
4.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
4.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
5 Forest2
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
5.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 31.78 1.40 29.01 34.46 1.00 1952 2187
## StageWithoutseedinflow 28.76 1.75 25.44 32.22 1.00 2030 1803
## StageWithseedinflow:Shannon -0.38 0.53 -1.39 0.67 1.00 1963 2089
## StageWithoutseedinflow:Shannon -6.10 0.95 -7.99 -4.27 1.00 2030 1795
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.61 0.27 10.10 11.17 1.00 2507 2207
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2941946 0.02337017 0.2477921 0.33987015.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
5.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
5.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
5.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
5.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 80.49 10.59 60.13 101.03 1.00 3606 2797
## Ninitial4:StageWithseedinflow 101.30 12.42 77.03 125.79 1.00 3302 2784
## Ninitial8:StageWithseedinflow 70.24 17.89 35.59 104.75 1.00 3028 2896
## Ninitial16:StageWithseedinflow 84.79 29.39 27.02 141.46 1.00 3167 2752
## Ninitial32:StageWithseedinflow 116.73 43.67 32.18 202.10 1.00 3602 2997
## Ninitial2:StageWithoutseedinflow 14.01 5.89 2.36 25.49 1.00 3333 2964
## Ninitial4:StageWithoutseedinflow 3.98 5.56 -7.22 15.05 1.00 3499 2921
## Ninitial8:StageWithoutseedinflow 15.66 6.09 3.83 27.53 1.00 3955 3035
## Ninitial16:StageWithoutseedinflow 8.04 5.51 -2.60 18.93 1.00 3544 2686
## Ninitial32:StageWithoutseedinflow 13.00 7.75 -2.37 28.23 1.00 3345 2665
## Ninitial2:StageWithseedinflow:Shannon -31.84 6.95 -45.35 -18.43 1.00 3605 2946
## Ninitial4:StageWithseedinflow:Shannon -33.52 5.87 -45.26 -22.11 1.00 3320 2731
## Ninitial8:StageWithseedinflow:Shannon -14.68 6.67 -27.59 -1.77 1.00 3048 2967
## Ninitial16:StageWithseedinflow:Shannon -16.27 8.78 -33.37 0.77 1.00 3176 2774
## Ninitial32:StageWithseedinflow:Shannon -21.81 11.11 -43.51 -0.30 1.00 3602 3009
## Ninitial2:StageWithoutseedinflow:Shannon 7.46 4.33 -1.14 15.98 1.00 3385 2859
## Ninitial4:StageWithoutseedinflow:Shannon 9.10 3.48 2.20 16.02 1.00 3482 2830
## Ninitial8:StageWithoutseedinflow:Shannon -0.06 3.04 -6.08 5.89 1.00 3940 2992
## Ninitial16:StageWithoutseedinflow:Shannon 2.49 2.59 -2.52 7.58 1.00 3557 2681
## Ninitial32:StageWithoutseedinflow:Shannon -0.10 3.12 -6.16 6.10 1.00 3332 2763
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 8.45 0.24 7.99 8.94 1.00 8198 2361
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4963703 0.01948423 0.4558494 0.53098535.2.1 Posterior predictive checks
5.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
5.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
5.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
6 Dryland
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
6.1 Across-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 73.89 1.28 71.32 76.35 1.00 2127 1917
## StageWithoutseedinflow 77.82 1.74 74.51 81.20 1.00 1993 2004
## StageWithseedinflow:Shannon 1.19 0.54 0.14 2.29 1.00 2179 2107
## StageWithoutseedinflow:Shannon -1.47 1.10 -3.61 0.69 1.00 1997 1822
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.38 0.24 8.91 9.86 1.00 2810 2355
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.01449498 0.007924524 0.00265956 0.033195146.1.1 Posterior predictive checks
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
6.1.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
6.1.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference. 
6.1.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 
6.2 Within-treatment effect
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 63.34 6.75 50.23 77.03 1.00 3706 2565
## Ninitial4:StageWithseedinflow 63.18 8.77 45.82 80.06 1.00 3872 2728
## Ninitial8:StageWithseedinflow 44.69 11.12 22.80 65.94 1.00 3940 2845
## Ninitial16:StageWithseedinflow 42.05 15.88 11.27 72.96 1.00 3771 3016
## Ninitial32:StageWithseedinflow 1.33 20.14 -38.02 41.82 1.00 3492 2704
## Ninitial2:StageWithoutseedinflow 73.36 4.28 65.02 81.88 1.00 3545 2892
## Ninitial4:StageWithoutseedinflow 60.21 4.44 51.18 68.75 1.00 3758 2649
## Ninitial8:StageWithoutseedinflow 55.08 4.57 46.12 64.18 1.00 3326 2920
## Ninitial16:StageWithoutseedinflow 41.80 6.52 28.50 54.69 1.00 3867 3055
## Ninitial32:StageWithoutseedinflow 38.07 9.53 19.45 56.82 1.00 4080 3063
## Ninitial2:StageWithseedinflow:Shannon 9.15 4.65 -0.36 18.26 1.00 3734 2749
## Ninitial4:StageWithseedinflow:Shannon 7.18 4.57 -1.69 16.28 1.00 3893 2639
## Ninitial8:StageWithseedinflow:Shannon 13.34 4.59 4.64 22.45 1.00 3949 3003
## Ninitial16:StageWithseedinflow:Shannon 11.69 5.36 1.35 22.15 1.00 3763 2897
## Ninitial32:StageWithseedinflow:Shannon 21.96 5.82 10.09 33.26 1.00 3471 2647
## Ninitial2:StageWithoutseedinflow:Shannon 5.38 3.42 -1.46 12.09 1.00 3518 2823
## Ninitial4:StageWithoutseedinflow:Shannon 11.38 3.04 5.61 17.52 1.00 3861 2607
## Ninitial8:StageWithoutseedinflow:Shannon 11.95 2.74 6.51 17.44 1.00 3350 2987
## Ninitial16:StageWithoutseedinflow:Shannon 16.44 3.49 9.59 23.57 1.00 3807 2914
## Ninitial32:StageWithoutseedinflow:Shannon 16.45 4.75 6.99 25.85 1.00 4084 2953
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 7.95 0.23 7.51 8.40 1.00 7185 2888
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2235789 0.02477866 0.174982 0.27114246.2.1 Posterior predictive checks
6.2.1.1 Density plot
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines). 
6.2.1.2 Scatter plot
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference. 
6.2.1.3 Highest-density interval
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information. 